Circulant decomposition of a matrix and the eigenvalues of Toeplitz type matrices

TL;DR

This paper introduces a novel circulant decomposition method for matrices, enabling efficient eigenvalue computation of Toeplitz and related matrices using FFT and improving preconditioning in linear solvers.

Contribution

It presents a new orthogonal decomposition of matrices into circulant components, facilitating fast eigenvalue approximation and preconditioning techniques.

Findings

Eigenvalues of Toeplitz matrices can be approximated efficiently.

Decomposition enables sparse similarity transformations with FFT.

Improves preconditioning in linear solvers.

Abstract

We begin by showing that any $n \times n$ matrix can be decomposed into a sum of $n$ circulant matrices with periodic relaxations on the unit circle. This decomposition is orthogonal with respect to a Frobenius inner product, allowing recursive iterations for these circulant components. It is also shown that the dominance of a few circulant components in the matrix allows sparse similarity transformations using Fast-Fourier-transform (FFT) operations. This enables the evaluation of all eigenvalues of dense Toeplitz, block-Toeplitz, and other periodic or quasi-periodic matrices, to a reasonable approximation in $\mathcal{O}(n^2)$ arithmetic operations. The utility of the approximate similarity transformation in preconditioning linear solvers is also demonstrated.